This solution is named after Willem Jacob van Stockum, who rediscovered it in 1938 independently of a much earlier discovery by Cornelius Lanczos in 1924.
That is, we demand that the world lines of the fluid particles form a timelike congruence having nonzero vorticity but vanishing expansion and shear.
(In fact, since dust particles feel no forces, this will turn out to be a timelike geodesic congruence, but we won't need to assume this in advance.)
A simple ansatz corresponding to this demand is expressed by the following frame field, which contains two undetermined functions of
: To prevent misunderstanding, we should emphasize that taking the dual coframe gives the metric tensor in terms of the same two undetermined functions: Multiplying out gives We compute the Einstein tensor with respect to this frame, in terms of the two undetermined functions, and demand that the result have the form appropriate for a perfect fluid solution with the timelike unit vector
Computing the Einstein tensor with respect to our frame shows that in fact the pressure vanishes, so we have a dust solution.
The mass density of the dust turns out to be Happily, this is finite on the axis of symmetry
, but the density increases with radius, a feature which unfortunately severely limits possible astrophysical applications.
vanishes, but the vorticity vector is This means that even though in our comoving chart the world lines of the dust particles appear as vertical lines, in fact they are twisting about one another as the dust particles swirl about the axis of symmetry.
In other words, if we follow the evolution of a small ball of dust, we find that it rotates about its own axis (parallel to
where q is a new undetermined function of r. Plugging in the requirement that the covariant derivatives vanish, we obtain The new frame appears, in our comoving coordinate chart, to be spinning, but in fact it is gyrostabilized.
This explains the physical meaning of the parameter which we found in our earlier derivation of the first frame.
(Pedantic note: alert readers will have noticed that we ignored the fact that neither of our frame fields is well defined on the axis.
This means that our on-axis observer sees the other dust particles at time-lagged locations, which is of course just what we would expect.
The fact that the null geodesics appear "bent" in this chart is of course an artifact of our choice of comoving coordinates in which the world lines of the dust particles appear as vertical coordinate lines.
Let us draw the light cones for some typical events in the van Stockum dust, to see how their appearance (in our comoving cylindrical chart) depends on the radial coordinate: As the figure[which?]
As we move further outward, we can see that horizontal circles with larger radii are closed timelike curves.
Even worse, there is apparently nothing to prevent such an observer from deciding, on his third lifetime, say, to stop accelerating, which would give him multiple biographies.
Closed timelike curves turn out to exist in many other exact solutions in general relativity, and their common appearance is one of the most troubling theoretical objections to this theory.